RLC Circuit Resonance: Damping, Ringing and the Tuning Dial

Three components, one second-order equation

A series RLC circuit — a resistor, inductor and capacitor in one loop — is governed by Kirchhoff's voltage law: the source voltage equals the sum of the drops across all three parts. Writing the drops in terms of the charge Q on the capacitor (current I = dQ/dt) gives a single linear second-order ordinary differential equation, exactly the same form as a damped mechanical oscillator:

L * d²Q/dt²  +  R * dQ/dt  +  Q/C  =  V(t)
         (inertia)        (damping)      (restoring force)

Natural frequency and damping ratio

Setting V(t) = 0 gives the free response. Two numbers control everything about its shape: the undamped natural (angular) frequency ω₀ and the damping ratio ζ:

ω₀ = 1 / sqrt(L * C)
ζ    = (R / 2) * sqrt(C / L)

When ζ < 1 the circuit is underdamped: charge and current oscillate at a frequency slightly below ω₀ while the envelope decays exponentially, ringing several times before settling. At ζ = 1 the circuit is critically damped: it returns to zero as fast as possible with no overshoot at all. When ζ > 1 it is overdamped: it crawls back to zero slowly, also without overshoot, because the resistive term dominates the dynamics and there is no oscillatory component left.

Driven resonance and impedance

Drive the circuit with a sinusoidal source at angular frequency ω instead, and the steady-state response is governed by the complex impedance:

Z(ω) = R + j * (ωL - 1/(ωC))
|Z(ω)| is minimised, and equal to R, exactly when ω = ω₀

At resonance the inductive and capacitive reactances cancel exactly, the circuit looks purely resistive, and the current amplitude reaches its maximum for a given drive voltage. How sharp that peak is depends on the quality factor Q = ω₀L/R = (1/R)√(L/C): a high-Q circuit has a narrow, tall resonance peak and a bandwidth of roughly ω₀/Q, while a low-Q circuit has a broad, shallow one. Equivalently, Q is (2π times) the ratio of energy stored in the circuit to energy dissipated per cycle — the less resistance bleeds energy away each cycle, the more cycles the ringing survives, and the sharper the resonance.

Why the same three letters show up everywhere

The RLC equation is mathematically identical to a mass on a spring with friction (L is inertia, 1/C is stiffness, R is friction), which is why intuition from one transfers directly to the other. Its most direct practical use is tuning: an old analogue radio's tuning dial physically changes C (or L) in an RLC circuit to move ω₀ until it lines up with the carrier frequency of the desired station, so that station's signal resonates strongly while every other frequency is suppressed — a band-pass filter built from three passive components and one resonance curve.

Frequently asked questions

Try it live

Everything above runs in your browser — open RLC Circuit Resonance and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.